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With their methods, we find a refined shift $\\sigma(t) = \\tilde \\sigma(t) + \\mu_* (\\log t)/t + \\alpha_1/t$ such that in the frame moving with $\\sigma$, the solution $u$ satisfies $u(t,x) = \\phi (x) + \\psi(x)/t + O(t^{\\gamma-3/2})$ for a certain profile $\\psi$ independent of"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1712.02472","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-12-07T02:08:58Z","cross_cats_sorted":[],"title_canon_sha256":"8d9bbe53a6fbc3abecaaceac80324cfa30f92b892b3c4128f59bb67e28e8e5e2","abstract_canon_sha256":"6eeadedc8e49c08d5d78a9010870ea601c21d818de462664ee3b233148af1710"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:28:34.017524Z","signature_b64":"vAyM0r6rDLMtafjkrl8bTJ3yumHCrnfJ55HS80+sp+Q4aTeQL0QP9dtl1HAVHEFQEN/YKGpO7/oYwCjrlNngBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ca43d4121576fcdeeffd3f6dfd0178320b7c2024099f92fb625028d07288899d","last_reissued_at":"2026-05-18T00:28:34.016727Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:28:34.016727Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Precise asymptotics for Fisher-KPP fronts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Cole Graham","submitted_at":"2017-12-07T02:08:58Z","abstract_excerpt":"We consider the one-dimensional Fisher-KPP equation with step-like initial data. 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